Complex Fourier analysis on a nilpotent Lie group

Goodman, Roe

doi:10.1090/S0002-9947-1971-0417334-3

Complex Fourier analysis on a nilpotent Lie group
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by Roe Goodman

Trans. Amer. Math. Soc. 160 (1971), 373-391

DOI: https://doi.org/10.1090/S0002-9947-1971-0417334-3

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Abstract:

Let $G$ be a simply-connected nilpotent Lie group, with complexification ${G_c}$. The functions on $G$ which are analytic vectors for the left regular representation of $G$ on ${L_2}(G)$ are determined in this paper, via a dual characterization in terms of their analytic continuation to ${G_c}$, and by properties of their ${L_2}$ Fourier transforms. The analytic continuation of these functions is shown to be given by the Fourier inversion formula. An explicit construction is given for a dense space of entire vectors for the left regular representation. In the case $G = R$ this furnishes a group-theoretic setting for results of Paley and Wiener concerning functions holomorphic in a strip.

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